Summary: Constant­Level Greedy Triangulations
Approximate the MWT Well
Oswin Aichholzer 1 , Franz Aurenhammer 1 , G¨unter Rote 2 , Yin­Feng Xu 3
1 Institute for Theoretical Computer Science, Graz University of Technology,
Klosterwiesgasse 32/2, A­8010 Graz, Austria
2 Institut f¨ur Mathematik, Technische Universit¨at Graz,
Steyrergasse 30, A­8010 Graz, Austria
3 School of Management, Xi'an Jiaotong University,
Xi'an Shaanxi, 710049, P. R. China
Abstract. The well­known greedy triangulation GT (S) of a finite point
set S is obtained by inserting compatible edges in increasing length order,
where an edge is compatible if it does not cross previously inserted ones.
Exploiting the concept of so­called light edges, we introduce a definition
of GT (S) that does not rely on the length ordering of the edges. Rather,
it provides a decomposition of GT (S) into levels, and the number of
levels allows us to bound the total edge length of GT (S). In particular,
we show jGT (S)j  3 \Delta 2 k+1 jMWT (S)j, where k is the number of levels
and MWT (S) is the minimum­weight triangulation of S.
1 Introduction
A triangulation of a given set S of n points in the plane is a maximal set of non­